A-LevelMathematicsDifferential equations9709/31 · Oct/Nov 2020 · Q86 Marks

The coordinates (x, y) of a general point of a curve satisfy the differential equation x dy/dx = (1 - 2x²)y, for x > 0. It is given that y = 1 when x = 1. Solve the differential equation, obtaining an expression for y in terms of x.

The coordinates (x, y) of a general point of a curve satisfy the differential equation x dy/dx = (1 - 2x²)y, for x > 0. It is given that y = 1 when x = 1. Solve the differential equation, obtaining an expression for y in terms of x.[6]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2020). The mark or mark type is shown in brackets.

Separate variables correctly and attempt integration of at least one side
Obtain term In y
Obtain terms Inx - x²
Use x = 1, y = 1 to evaluate a constant, or as limits, in a solution containing at least 2 terms of the form alny, blnx and cx²
Obtain correct solution in any form
Rearrange and obtain y = xe^(1-x²)

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Topic: Differential equations, A-Level Mathematics (syllabus 9709). Structured question, 6 marks.
Source: A-Level Mathematics, Oct/Nov 2020 Paper 3 Variant 1, question 8.

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